Question: $g(x)=(x+9)(x+8)$ 1) What are the zeros of the function? Write the smaller $x$ first, and the larger $x$ second. $\text{smaller }x=$
Solution: $\begin{aligned} (x+9)(x+8)&=0 \\\\ x+9=0&\text{ or }x+8=0 \\\\ x={-9}&\text{ or }x={-8} \end{aligned}$ There are many ways to find the vertex. We will do it by using the fact that the $x$ -coordinate of the vertex is exactly between the two zeros. $\begin{aligned} \text{vertex's }x\text{-coordinate}&=\dfrac{({-9})+({-8})}{2} \\\\ &={-\dfrac{17}{2}} \end{aligned}$ Now we can find the vertex's $y$ -coordinate by evaluating $g\left({-\dfrac{17}{2}}\right)$ : $\begin{aligned} g\left({-\dfrac{17}{2}}\right)&=\left({-\dfrac{17}{2}}+9\right)\left({-\dfrac{17}{2}}+8\right) \\\\ &=\left(\dfrac12\right)\left(-\dfrac12\right) \\\\ &=-\dfrac{1}{4} \end{aligned}$ In conclusion, $\begin{aligned} \text{smaller }x&=-9 \\\\ \text{larger }x&=-8 \end{aligned}$ The vertex of the parabola is at $\left(-\dfrac{17}{2},-\dfrac{1}{4}\right)$